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Theorem rankelop 8283
Description: Rank membership is inherited by ordered pairs. (Contributed by NM, 18-Sep-2006.)
Hypotheses
Ref Expression
rankelun.1  |-  A  e. 
_V
rankelun.2  |-  B  e. 
_V
rankelun.3  |-  C  e. 
_V
rankelun.4  |-  D  e. 
_V
Assertion
Ref Expression
rankelop  |-  ( ( ( rank `  A
)  e.  ( rank `  C )  /\  ( rank `  B )  e.  ( rank `  D
) )  ->  ( rank `  <. A ,  B >. )  e.  ( rank `  <. C ,  D >. ) )

Proof of Theorem rankelop
StepHypRef Expression
1 rankelun.1 . . . 4  |-  A  e. 
_V
2 rankelun.2 . . . 4  |-  B  e. 
_V
3 rankelun.3 . . . 4  |-  C  e. 
_V
4 rankelun.4 . . . 4  |-  D  e. 
_V
51, 2, 3, 4rankelpr 8282 . . 3  |-  ( ( ( rank `  A
)  e.  ( rank `  C )  /\  ( rank `  B )  e.  ( rank `  D
) )  ->  ( rank `  { A ,  B } )  e.  (
rank `  { C ,  D } ) )
6 rankon 8204 . . . . 5  |-  ( rank `  { C ,  D } )  e.  On
76onordi 4971 . . . 4  |-  Ord  ( rank `  { C ,  D } )
8 ordsucelsuc 6630 . . . 4  |-  ( Ord  ( rank `  { C ,  D }
)  ->  ( ( rank `  { A ,  B } )  e.  (
rank `  { C ,  D } )  <->  suc  ( rank `  { A ,  B } )  e.  suc  ( rank `  { C ,  D } ) ) )
97, 8ax-mp 5 . . 3  |-  ( (
rank `  { A ,  B } )  e.  ( rank `  { C ,  D }
)  <->  suc  ( rank `  { A ,  B }
)  e.  suc  ( rank `  { C ,  D } ) )
105, 9sylib 196 . 2  |-  ( ( ( rank `  A
)  e.  ( rank `  C )  /\  ( rank `  B )  e.  ( rank `  D
) )  ->  suc  ( rank `  { A ,  B } )  e. 
suc  ( rank `  { C ,  D }
) )
111, 2rankop 8267 . . 3  |-  ( rank `  <. A ,  B >. )  =  suc  suc  ( ( rank `  A
)  u.  ( rank `  B ) )
121, 2rankpr 8266 . . . 4  |-  ( rank `  { A ,  B } )  =  suc  ( ( rank `  A
)  u.  ( rank `  B ) )
13 suceq 4932 . . . 4  |-  ( (
rank `  { A ,  B } )  =  suc  ( ( rank `  A )  u.  ( rank `  B ) )  ->  suc  ( rank `  { A ,  B } )  =  suc  suc  ( ( rank `  A
)  u.  ( rank `  B ) ) )
1412, 13ax-mp 5 . . 3  |-  suc  ( rank `  { A ,  B } )  =  suc  suc  ( ( rank `  A
)  u.  ( rank `  B ) )
1511, 14eqtr4i 2486 . 2  |-  ( rank `  <. A ,  B >. )  =  suc  ( rank `  { A ,  B } )
163, 4rankop 8267 . . 3  |-  ( rank `  <. C ,  D >. )  =  suc  suc  ( ( rank `  C
)  u.  ( rank `  D ) )
173, 4rankpr 8266 . . . 4  |-  ( rank `  { C ,  D } )  =  suc  ( ( rank `  C
)  u.  ( rank `  D ) )
18 suceq 4932 . . . 4  |-  ( (
rank `  { C ,  D } )  =  suc  ( ( rank `  C )  u.  ( rank `  D ) )  ->  suc  ( rank `  { C ,  D } )  =  suc  suc  ( ( rank `  C
)  u.  ( rank `  D ) ) )
1917, 18ax-mp 5 . . 3  |-  suc  ( rank `  { C ,  D } )  =  suc  suc  ( ( rank `  C
)  u.  ( rank `  D ) )
2016, 19eqtr4i 2486 . 2  |-  ( rank `  <. C ,  D >. )  =  suc  ( rank `  { C ,  D } )
2110, 15, 203eltr4g 2560 1  |-  ( ( ( rank `  A
)  e.  ( rank `  C )  /\  ( rank `  B )  e.  ( rank `  D
) )  ->  ( rank `  <. A ,  B >. )  e.  ( rank `  <. C ,  D >. ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1398    e. wcel 1823   _Vcvv 3106    u. cun 3459   {cpr 4018   <.cop 4022   Ord word 4866   suc csuc 4869   ` cfv 5570   rankcrnk 8172
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-reg 8010  ax-inf2 8049
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-om 6674  df-recs 7034  df-rdg 7068  df-r1 8173  df-rank 8174
This theorem is referenced by: (None)
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